Chapter 4.4, Problem LC

LINKING concepts... For Individual or Group Explorations

The Logistic Growth Model

When a virus infects a finite population of size P in which no one is immune, the virus spreads slowly at first, then rapidly like exponential growth. Since the population is finite, the exponential growth must slow down as nearly everyone becomes infected. This type of behavior was named the logistic function or logistic curve in 1844 by Pierre Franҫ? ois Verhulst. The logistic function has the form

$\langle \text{em}\rangle \langle p\rangle n=\frac{p}{1+\left(p\&\#8722;-1\right)\&\#183;·e^{\&\#8722;-ct}}\langle /p\rangle \langle /\text{em}\rangle$ where n is the number of people who have caught the virus on or before day t, and c is a positive constant.

a) According to the model, how many people have caught the virus at time t = 0?

b) Now consider what happens when one student carrying a flu virus returns from spring break to a university of 10,000 students. For c = 0.1,0.2, and so on through c = 0.9, graph the logistic curve

y₁ = 10000/ (1 + 9999e^(–cx))

and the daily number of new cases

y₂= y₁(x) – y₁(x – 1)

For each value of c use the graph of y₂ to find the day on which the flu is spreading most rapidly.

c) The Health Center estimated that the greatest number of new cases of the flu occurred on the 19th day after the students returned from spring break. What value of c should be used to model this situation? How many new cases occurred on the 19th day?

d) Algebraically find the day on which the number of infected students reached 9000.

e) The Health Center has a team of doctors from Atlanta arriving on the 30th day to help with this three-day flu epidemic. What do you think of this plan?